ISC Class 12 Mathematics Competency Focused Questions 2026, Download PDF for Board Exam Preparation

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Details

Article Title

ISC Class 12 Mathematics Competency Focused Questions 2026

Conducting Body

Council for the Indian School Certificate Examinations (CISCE)

Exam Level

ISC Class 12

Subject

Mathematics

Paper

Theory Paper

Exam Date 2026

Monday, March 9, 2026

Exam Duration

3 Hours

Maximum Marks

80 Marks (External Assessment)

Related Resource

ISC Class 12 Mathematics Specimen Paper 2026

Official Website

cisce.org 

Question Number

Questions

1

$$ \text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} $$
$$
\begin{array}{l}
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

$$
\begin{array}{l}
\text{(a) } \sin \left(\cos^{-1} x\right) = \cos \left(\sin^{-1} x\right) \\
\text{(b) } \sec \left(\tan^{-1} x\right) = \tan \left(\sec^{-1} x\right) \\
\text{(c) } \cos \left(\tan^{-1} x\right) = \tan \left(\cos^{-1} x\right) \\
\text{(d) } \tan \left(\sin^{-1} x\right) = \sin \left(\tan^{-1} x\right)
\end{array}
$$

$$ \text{If a matrix } A = \left[a_{ij}\right]_{2\times2}, \text{ where } a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}, \text{ then } A^{-1} \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } 5 \\
\text{(b) } \frac{1}{5} \\
\text{(c) } 125 \\
\text{(d) } 25
\end{array}
$$

$$ \text{If } h(x) = 4^{x} \text{ and } h^{-1}(x) = 2, \text{ then value of } x \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } -4 \\
\text{(b) } 4 \\
\text{(c) } -16 \\
\text{(d) } 16
\end{array}
$$

2

$$
\begin{array}{l}
\text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} \\
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

3

Which one of the following is true?

$$
\begin{array}{l}
\text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} \\
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

$$
\begin{array}{l}
\text{(a) } \sin (\cos^{-1} x) = \cos (\sin^{-1} x) \\
\text{(b) } \sec (\tan^{-1} x) = \tan (\sec^{-1} x) \\
\text{(c) } \cos (\tan^{-1} x) = \tan (\cos^{-1} x) \\
\text{(d) } \tan (\sin^{-1} x) = \sin (\tan^{-1} x)
\end{array}
$$

4

$$ \text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} $$
$$
\begin{array}{l}
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

$$
\begin{array}{l}
\text{(a) } \sin \left(\cos^{-1} x\right) = \cos \left(\sin^{-1} x\right) \\
\text{(b) } \sec \left(\tan^{-1} x\right) = \tan \left(\sec^{-1} x\right) \\
\text{(c) } \cos \left(\tan^{-1} x\right) = \tan \left(\cos^{-1} x\right) \\
\text{(d) } \tan \left(\sin^{-1} x\right) = \sin \left(\tan^{-1} x\right)
\end{array}
$$

$$ \text{If a matrix } A = \left[a_{ij}\right]_{2\times2}, \text{ where } a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}, \text{ then } A^{-1} \text{ is:} $$

(a) I
(b) A
(c) − AA
(d) − I

5

$$ \text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} $$
$$
\begin{array}{l}
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

$$
\begin{array}{l}
\text{(a) } \sin \left(\cos^{-1} x\right) = \cos \left(\sin^{-1} x\right) \\
\text{(b) } \sec \left(\tan^{-1} x\right) = \tan \left(\sec^{-1} x\right) \\
\text{(c) } \cos \left(\tan^{-1} x\right) = \tan \left(\cos^{-1} x\right) \\
\text{(d) } \tan \left(\sin^{-1} x\right) = \sin \left(\tan^{-1} x\right)
\end{array}
$$

$$ \text{If a matrix } A = \left[a_{ij}\right]_{2\times2}, \text{ where } a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}, \text{ then } A^{-1} \text{ is:} $$

$$ \text{If } h(x) = 4^{x} \text{ and } h^{-1}(x) = 2, \text{ then value of } x \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } -4 \\
\text{(b) } 4 \\
\text{(c) } -16 \\
\text{(d) } 16
\end{array}
$$

$$ \text{If the value of } 3^{\text{rd}} \text{ order determinant is 5, then the value of determinant formed by replacing} $$
$$ \text{its element by its co-factor is:} $$
$$
\begin{array}{l}
\text{(a) } 5 \\
\text{(b) } \frac{1}{5} \\
\text{(c) } 125 \\
\text{(d) } 25
\end{array}
$$

6

$$ \text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} $$
$$
\begin{array}{l}
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

$$
\begin{array}{l}
\text{(a) } \sin \left(\cos^{-1} x\right) = \cos \left(\sin^{-1} x\right) \\
\text{(b) } \sec \left(\tan^{-1} x\right) = \tan \left(\sec^{-1} x\right) \\
\text{(c) } \cos \left(\tan^{-1} x\right) = \tan \left(\cos^{-1} x\right) \\
\text{(d) } \tan \left(\sin^{-1} x\right) = \sin \left(\tan^{-1} x\right)
\end{array}
$$

$$ \text{If a matrix } A = \left[a_{ij}\right]_{2\times2}, \text{ where } a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}, \text{ then } A^{-1} \text{ is:} $$

$$ \text{If the value of } 3^{\text{rd}} \text{ order determinant is 5, then the value of determinant formed by replacing} $$
$$ \text{its element by its co-factor is:} $$
$$
\begin{array}{l}
\text{(a) } 5 \\
\text{(b) } \frac{1}{5} \\
\text{(c) } 125 \\
\text{(d) } 25
\end{array}
$$

$$ \text{If } h(x) = 4^{x} \text{ and } h^{-1}(x) = 2, \text{ then value of } x \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } -4 \\
\text{(b) } 4 \\
\text{(c) } -16 \\
\text{(d) } 16
\end{array}
$$

$$ \text{If } A = \begin{bmatrix} 0 & 5 & -y \\ -5 & 0 & x \\ y & -x & 0 \end{bmatrix}, \text{ then the value of } A^{-1} \cdot (\text{adj } A)A \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } A^2 \\
\text{(b) } I \\
\text{(c) } 0 \\
\text{(d) } A
\end{array}
$$

7

$$ \text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} $$
$$
\begin{array}{l}
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

$$
\begin{array}{l}
\text{(a) } \sin \left(\cos^{-1} x\right) = \cos \left(\sin^{-1} x\right) \\
\text{(b) } \sec \left(\tan^{-1} x\right) = \tan \left(\sec^{-1} x\right) \\
\text{(c) } \cos \left(\tan^{-1} x\right) = \tan \left(\cos^{-1} x\right) \\
\text{(d) } \tan \left(\sin^{-1} x\right) = \sin \left(\tan^{-1} x\right)
\end{array}
$$

$$ \text{If a matrix } A = \left[a_{ij}\right]_{2\times2}, \text{ where } a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}, \text{ then } A^{-1} \text{ is:} $$

$$ \text{If the value of } 3^{\text{rd}} \text{ order determinant is 5, then the value of determinant formed by replacing} $$
$$ \text{its element by its co-factor is:} $$
$$
\begin{array}{l}
\text{(a) } 5 \\
\text{(b) } \frac{1}{5} \\
\text{(c) } 125 \\
\text{(d) } 25
\end{array}
$$

$$ \text{If } h(x) = 4^{x} \text{ and } h^{-1}(x) = 2, \text{ then value of } x \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } -4 \\
\text{(b) } 4 \\
\text{(c) } -16 \\
\text{(d) } 16
\end{array}
$$

$$ \text{If } A = \begin{bmatrix} 0 & 5 & -y \\ -5 & 0 & x \\ y & -x & 0 \end{bmatrix}, \text{ then the value of } A^{-1} \cdot (\text{adj } A)A \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } A^2 \\
\text{(b) } I \\
\text{(c) } 0 \\
\text{(d) } A
\end{array}
$$

$$ \text{If } D = \begin{vmatrix} p & p & p \\ p & p + x & p \\ p & p & p + y \end{vmatrix} \text{ for } p \neq 0, x \neq 0, y \neq 0 \text{ then D is divisible by:} $$
$$
\begin{array}{l}
\text{(a) only p.} \\
\text{(b) p and x but not y.} \\
\text{(c) p and y but not x.} \\
\text{(d) p, x and y.}
\end{array}
$$

8

$$ \text{If } a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b, \text{ then } a \text{ and } b \text{ are respectively:} $$
$$
\begin{array}{l}
\text{(a) } \frac{\pi}{2} \text{ and } 2\pi \\
\text{(b) } \frac{\pi}{2} \text{ and } -\frac{\pi}{2} \\
\text{(c) } 0 \text{ and } \pi \\
\text{(d) } 0 \text{ and } 2\pi
\end{array}
$$

$$
\begin{array}{l}
\text{(a) } \sin \left(\cos^{-1} x\right) = \cos \left(\sin^{-1} x\right) \\
\text{(b) } \sec \left(\tan^{-1} x\right) = \tan \left(\sec^{-1} x\right) \\
\text{(c) } \cos \left(\tan^{-1} x\right) = \tan \left(\cos^{-1} x\right) \\
\text{(d) } \tan \left(\sin^{-1} x\right) = \sin \left(\tan^{-1} x\right)
\end{array}
$$

$$ \text{If a matrix } A = \left[a_{ij}\right]_{2\times2}, \text{ where } a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}, \text{ then } A^{-1} \text{ is:} $$

$$ \text{If the value of } 3^{\text{rd}} \text{ order determinant is 5, then the value of determinant formed by replacing} $$
$$ \text{its element by its co-factor is:} $$
$$
\begin{array}{l}
\text{(a) } 5 \\
\text{(b) } \frac{1}{5} \\
\text{(c) } 125 \\
\text{(d) } 25
\end{array}
$$

$$ \text{If } h(x) = 4^{x} \text{ and } h^{-1}(x) = 2, \text{ then value of } x \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } -4 \\
\text{(b) } 4 \\
\text{(c) } -16 \\
\text{(d) } 16
\end{array}
$$

$$ \text{If } A = \begin{bmatrix} 0 & 5 & -y \\ -5 & 0 & x \\ y & -x & 0 \end{bmatrix}, \text{ then the value of } A^{-1} \cdot (\text{adj } A)A \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } A^2 \\
\text{(b) } I \\
\text{(c) } 0 \\
\text{(d) } A
\end{array}
$$

$$ \text{If } D = \begin{vmatrix} p & p & p \\ p & p + x & p \\ p & p & p + y \end{vmatrix} \text{ for } p \neq 0, x \neq 0, y \neq 0 \text{ then D is divisible by:} $$
$$
\begin{array}{l}
\text{(a) only p.} \\
\text{(b) p and x but not y.} \\
\text{(c) p and y but not x.} \\
\text{(d) p, x and y.}
\end{array}
$$

$$ \text{If adj}(A) = \begin{bmatrix} 2 & 3 & 5 \\ x & 5 & 1 \\ 3 & 3 & 4 \end{bmatrix} \text{ and } |A| = 4, \text{ then the value of } x \text{ is:} $$
$$
\begin{array}{l}
\text{(a) } 16 \\
\text{(b) } 12 \\
\text{(c) } 32 \\
\text{(d) } 10
\end{array}
$$

ISC Class 12 Mathematics Competency Focused Questions 2026 Download PDF

Monday, March 9

Mathematics